Improved estimate of the singular set of Dir-minimizing Q-valued functions via an abstract regularity result

In this note we prove an abstract version of a recent quantitative stratification principle introduced by Cheeger and Naber (2013) [6,7]. Using this general regularity result paired with an ε-regularity theorem we provide a new estimate of the Minkowski dimension of the set of higher multiplicity points of a Dir-minimizing Q-valued function. The abstract principle is applicable to several other problems: we recover recent results in the literature and we obtain also some improvements in more classical contexts.